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Mathematics · Class 9 · Chapter 5: Algebraic Manipulation

8 min

Key points

Algebraic expressions consist of constants, variables and algebraic operations
Like terms have the same variables raised to the same powers
Polynomial is an algebraic expression in which variables have non-negative integral powers
Degree of a polynomial is the highest power of the variable in the expression
Binomial has exactly two terms, trinomial has exactly three terms
Addition and subtraction of polynomials is done by combining like terms
Multiplication of polynomials follows distributive property
Factor theorem states: if f(a) = 0, then (x-a) is a factor of f(x)
Remainder theorem states: when polynomial f(x) is divided by (x-a), remainder is f(a)

Definitions

Polynomial: An algebraic expression in which the variables have non-negative integral powers and no variable appears in the denominator.
Degree of Polynomial: The highest power of the variable in a polynomial is called the degree of the polynomial.
Like Terms: The terms having the same variables raised to the same powers are called like terms.
Factor Theorem: If f(x) is a polynomial and f(a) = 0, then (x-a) is a factor of f(x).
Remainder Theorem: When a polynomial f(x) is divided by (x-a), the remainder is f(a).

Worked example

Factorize the expression: x² + 7x + 12 (4 marks)

x² + 7x + 12 = (x + 3)(x + 4)

Common mistakes

Students often confuse coefficient with degree of polynomial
Forgetting to write f(a) = 0 condition when applying factor theorem
Not combining like terms properly during addition/subtraction
Making sign errors when multiplying polynomials with negative terms
Confusing remainder theorem with factor theorem

Quick recap

Algebraic manipulation involves expanding brackets using distributive law, factorizing by finding common factors or factor pairs, and simplifying by collecting like terms. Always show step-by-step working for full marks. Remember BODMAS order when simplifying complex expressions. Factorization is the reverse of expansion.

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